A circle with circumference $12\pi$ has an arc with a $45^\circ$ central angle. What is the length of the arc? ${12\pi}$ ${45^\circ}$ $\color{#DF0030}{\dfrac{3}{2}\pi}$
Solution: The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{45^\circ}{360^\circ} = \dfrac{s}{12\pi}$ $\dfrac{1}{8} = \dfrac{s}{12\pi}$ $\dfrac{1}{8} \times 12\pi = s$ $\dfrac{3}{2}\pi = s$